The special values of Dirichlet L-series

In this talk we will give an introduction to the Bloch-Kato conjecture on special values for Dirichlet L-series. For a Dirichlet character χ : ℤ/Nℤ → ℂ^{×} one defines the L-function (for Re(s) > 1)

L(χ, s) = ∑_{k=1}^{∞} \frac{χ(k)}{k^{s}}

where χ(k) = 0 if (k, N) ≠ 1. This has a meromorphic continuation to ℂ. The Bloch-Kato conjecture gives a formula for the leading coefficient L(χ, n)^{∗} of the Taylor series expansion of L(χ, s) at n ∈ ℤ in terms of certain p-adic and ∞-regulators. At the end of the talk we will very briefly explain the proof of the following theorem, which is joint work with Annette Huber: Theorem: The Bloch-Kato conjecture is true for L(χ, n)^{∗} for all n ∈ ℤ.